Since January 1st 2009, EU citizens do not need a work permit in Hungary.
As a non-EU citizen, if you are a spouse of a Hungarian citizen, you do not need a work permit.

All non-EU citizens require a work visa and permit to be able to enter and then work legally. Normally, most of the bureaucracy is handled by the employer. The hiring company/school must advertise the job at the Hungarian Labour Office ( Munkaugyi Kozpont) for 15 days. This gives a chance for unemployed Hungarians to apply.

After being granted a work permit, the employee has to apply for the working visa in person at the Hungarian Consulate in his/her home country. This will take about 25 days minimum in total, but the visa would then be for 2 years.

Obviously, it is much easier to hire an EU citizen, so as a non-EU citizen you would need some special skills/qualifications in order for the employer to go to such trouble for you. However, it is not impossible.